Generalizations of the Muller-Schupp theorem and tree-like inverse graphs

We extend the characterization of context-free groups of Muller and Schupp in two ways. We first show that for a quasi-transitive inverse graph Gamma$\Gamma$, being quasi-isometric to a tree, or context-free in the sense of Muller-Schupp (finitely many end-cone up to end-isomorphism), or having the automorphism group Aut(Gamma)$\operatorname{Aut}(\Gamma)$ that is virtually free, are all equivalent conditions. Furthermore, we add to the previous equivalences a group theoretic analog to the representation theorem of Chomsky-Sch & uuml;tzenberger that is fundamental in solving a weaker version of a conjecture of Brough which also extends Muller and Schupp's result to the class of groups that are virtually finitely generated subgroups of the direct product of free groups. We show that such groups are precisely those whose word problem is the intersection of a finite number of languages accepted by quasi-transitive, tree-like inverse graphs.